Geometric localization in supported elastic struts

Elastic fingering in a rotating Hele-Shaw cell

We consider the steady-state fingering instability of an elastic membrane separating two fluids of different density under external pressure in a rotating Hele-Shaw cell. Both inextensible and highly extensible membranes are considered, and the role of membrane tension is detailed in each case. Both systems exhibit a centrifugally driven Rayleigh-Taylor-like instability when the density of the inner fluid exceeds that of the outer one, and this instability competes with the restoring forces arising from curvature and tension, thereby setting the finger scale. Numerical continuation is used to compute not only strongly nonlinear primary finger states up to the point of self-contact, but also secondary branches of mixed modes and circumferentially localized folds as a function of the rotation rate and the externally imposed pressure. Both reflection-symmetric and symmetry-broken chiral states are computed. The results are presented in the form of bifurcation diagrams. The ratio of system scale to the natural length scale is found to determine the ordering of the primary bifurcations from the unperturbed circle state as well as the solution profiles and onset of secondary bifurcations.

Self-organized rod undulations on pre-stretched textiles

Textile technology is a traditional approach to additive manufacturing based on one-dimensional yarn. Printing solid rods onto pre-stretched textiles creates internal stresses upon relaxation of the pre-stretch, which leads to buckling-induced out-of-plane deformation of the textile. Similar behaviours are well known to occur also in biological systems where differential growth leads to internal stresses that are responsible for the folding or wrinkling of leaves, for example. Our goal was to get a quantitative understanding of this wrinkling by a systematic experimental and numerical investigation of parallel rods printed onto a pre-stretched textile. We vary rod thickness and spacing to obtain wavelength and phase coherence of the wrinkles as a function of these parameters. We also derive a simple analytical description to rationalize these observations. The result is a simple analytical estimate for the phase diagram of behaviours that may be used for design purposes or to describe wrinkling phenomena in biological or bioinspired systems.

Puckering and wrinkling in a growing composite ring

Pattern formation driven by differential strain in constrained elastic systems is a common motif in many technological and biological systems. Here we introduce a biologically motivated case of elastic patterning that allows us to explore the conditions for the existence of local puckering and global wrinkling patterns: a soft growing composite ring adhered elastically to a constraining rigid ring. We explore how differential growth of the soft ring and the elastic resistance to shear and stretching deformations induced by soft adherence lead to a range of phenomena that include uniform aperture-like modes, localized puckers that are Nambu–Goldstone-like modes and global wrinkles in the system. Our analysis combines computer simulations of a discrete rod model with a nonlinear stability analysis of the differential equations in the continuum limit. We provide phase diagrams and scaling relations that reveal the nature and extent of the deformation patterns. Overall, our study reveals how geometry and mechanics conspire to yield a rich phenomenology that could serve as a guide to the design of programmable localized elastic deformations while being relevant for the mechanical basis of biological morphogenesis.